A101703 Numbers n such that reversal(n) = (2/3)*n - 2.

21, 291, 885, 2991, 29991, 234651, 299991, 2340651, 2999991, 8221845, 23400651, 29346591, 29999991, 234000651, 293406591, 299999991, 2340000651, 2346534651, 2934006591, 2993465991, 2999999991, 23400000651, 23465934651, 29340006591, 29934065991, 29999999991, 82277815845, 234000000651

Offset: 1

Author

Farideh Firoozbakht, Dec 31 2004

Keywords

• base
• nonn

Comments

Numbers of the form 3*(10^n-3) are in the sequence, so A086947 is an infinite subsequence of this sequence. Also A101700 is a subsequence of this sequence.

Let f(r,s,t,z) = 2.(9)(r+s).(34.(0)(t).65)(z).(9)(s).1 where the dot between numbers means concatenation and "(m)(n)" means number of m's is n, for example f(0,2,1,3)= 299340653406534065991, it is interesting that all numbers of the form f(r,s,t,z) where r, s, t & z are nonnegative integers and r*z=0 are in this sequence.

Except for 885 & 8221845 all known terms of this sequence are of the form f(r,s,t,z).

For all r, s & t we have f(r,s,t,0)=f(r,s,0,0)=f(r+2s,0,0,0)=A086947(r+2s+1)= 3*(10^(r+2s+1)-3).

a(1) = 21 = f(0,0,0,0), a(2) = 291 = f(1,0,0,0), a(4) = 2991 = f(2,0,0,0) = f(0,1,0,0), a(5) = 29991 = f(3,0,0,0) = f(1,1,0,0), a(6) = 234651 = f(0,0,0,1), a(7) = 299991 = f(4,0,0,0) = f(0,2,0,0), a(8) = 2340651 = f(0,0,1,1), etc. Next term is greater than 11*10^8.

From David Wasserman, Mar 27 2008: (Start)

234653406534651 is a term that doesn't fit the f(r,s,t,z) format.

We may redefine f so that t is a vector of length z, which must be symmetrical to produce a member. For example f(0,0,[0,1,0],3) = 234653406534651 is a member, but f(0,0,[1,0,0],3) = 234065346534651 is not a member.

23465934651 is another member that doesn't fit the pattern. In general there may be any number of 9's between a 5 and a 3, provided that the 9's are symmetrical. So 2346593465934651 is a member, but 23465993465934651 is not. (End)

Examples

			f(0,1,2,3) = 2934006534006534006591 is in the sequence because reversal(2934006534006534006591) = 1956004356004356004392 = (2/3)*2934006534006534006591-2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..56 (terms < 10^14)

Crossrefs

Cf. A086947, A069215, A101700, A101704.

Programs

Mathematica

Do[If[FromDigits[Reverse[IntegerDigits[n]]] == 2/3*n - 2, Print[n]], {n, 1100000000}]

Extensions

More terms from David Wasserman, Mar 27 2008